Scaling behavior of linear polymers in disordered media

It has long been known that the universal scaling properties of linear polymers in disordered media are well described by the statistics of self-avoiding walks (SAWs) on percolation clusters and their critical exponent ${\nu }_{\mathrm{SAW}}$, with the SAW implicitly referring to the average SAW. Hitherto, static averaging has been commonly used, e.g., in numerical simulations, to determine what the average SAW is. We assert that only kinetic, rather than static, averaging can lead to asymptotic scaling behavior and corroborate our assertion by heuristic arguments and a renormalizable field theory. Moreover, we calculate to two-loop order ${\nu }_{\mathrm{SAW}}$, the exponent ${\nu }_{\max }$ for the longest SAW, and a family of multifractal exponents ${\nu }^{\left(\alpha \right)}$.

Figure 1

Percolation cluster in the node-link-blob picture.

Figure 2

(Color online) Static and kinetic rule for averaging SAWs.

Figure 3

(Color online) The $\epsilon$ expansions of the exponents ${\nu }_{\min }$ (blue top curve), ${\nu }_{\mathrm{SAW}}$ (red middle curve), and ${\nu }_{\max }$ (green bottom curve). Possible extrapolations at low dimensions $d$ are shown by dashed lines. The squares with error bars symbolize numerical results for ${\nu }_{\mathrm{SAW}}$ as compiled in [1].